Did Juror #3 finally believe the boy was not guilty, or did he vote just to get it over with?

I think that, in the end, even Juror #3 was convinced of reasonable doubt. 

He seems almost to give in at the end simply because everyone else has changed their votes.  When the others realize that some of his disdain for the boy is based on the fact that his relationship with his own son is dysfunctional, he simply folds under the pressure.  He does not want to give in, but he has no real reason to fight against the rest of the jury once they have all changed their votes.

I think that Juror #3 voted not guilty just to get over it because he couldn’t stand alone and he no more reasons or facts to argue with the rest of the jurors.

I believe that Three simply just caves in due to peer pressure at the end because throughout the entire play/story Three has always been the main antagonist trying to put down any of Eight or the other jurors' thoughts if they were related to not guilty. In the end, we can see him desperately pleading to Four to change his vote and “come back,” at this point we know and he knows himself that he had run out of time, supporters, and ideas to support himself in the argument. When the others are jeering at him and he says, “(shouting). Well, you're not going to intimidate me! (They all look at NO. 3.) I'm entitled to my opinion! (No one answers him.) It's gonna be a hung jury! That's it!” It is more to himself as comfort to himself and his last thread of hope and bias hanging on to his verdict. Everyone in the jury knows that Three is stubborn and as the reader knows, Three is coming from a place of extreme bias because of his past relations, so at the end Three has not learned to let go due to pure stubbornness and just caves in because he doesn't have the mental fortitude to let go of the past.